3.2884 \(\int \frac{1}{(c e+d e x)^2 \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=166 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} d e^2}-\frac{1}{a d e^2 (c+d x)} \]

[Out]

-(1/(a*d*e^2*(c + d*x))) + (b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt
[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*d*e^2) + (b^(1/3)*Log[a^(1/3) + b^(1/3)*(c + d*x
)])/(3*a^(4/3)*d*e^2) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/
3)*(c + d*x)^2])/(6*a^(4/3)*d*e^2)

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Rubi [A]  time = 0.328423, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d e^2}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d e^2}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} d e^2}-\frac{1}{a d e^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)^2*(a + b*(c + d*x)^3)),x]

[Out]

-(1/(a*d*e^2*(c + d*x))) + (b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt
[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*d*e^2) + (b^(1/3)*Log[a^(1/3) + b^(1/3)*(c + d*x
)])/(3*a^(4/3)*d*e^2) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/
3)*(c + d*x)^2])/(6*a^(4/3)*d*e^2)

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Rubi in Sympy [A]  time = 40.1681, size = 158, normalized size = 0.95 \[ - \frac{1}{a d e^{2} \left (c + d x\right )} + \frac{\sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{4}{3}} d e^{2}} - \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{4}{3}} d e^{2}} + \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{4}{3}} d e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*e*x+c*e)**2/(a+b*(d*x+c)**3),x)

[Out]

-1/(a*d*e**2*(c + d*x)) + b**(1/3)*log(a**(1/3) + b**(1/3)*(c + d*x))/(3*a**(4/3
)*d*e**2) - b**(1/3)*log(a**(2/3) + a**(1/3)*b**(1/3)*(-c - d*x) + b**(2/3)*(c +
d*x)**2)/(6*a**(4/3)*d*e**2) + sqrt(3)*b**(1/3)*atan(sqrt(3)*(a**(1/3)/3 + b**(
1/3)*(-2*c/3 - 2*d*x/3))/a**(1/3))/(3*a**(4/3)*d*e**2)

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Mathematica [A]  time = 0.0817886, size = 143, normalized size = 0.86 \[ \frac{-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{6 \sqrt [3]{a}}{c+d x}}{6 a^{4/3} d e^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)^2*(a + b*(c + d*x)^3)),x]

[Out]

((-6*a^(1/3))/(c + d*x) - 2*Sqrt[3]*b^(1/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*(c + d*
x))/(Sqrt[3]*a^(1/3))] + 2*b^(1/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)] - b^(1/3)*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/(6*a^(4/3)*d*e^2)

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Maple [C]  time = 0.006, size = 98, normalized size = 0.6 \[ -{\frac{1}{3\,{e}^{2}ad}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{{e}^{2}ad \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)^2/(a+b*(d*x+c)^3),x)

[Out]

-1/3/e^2/a/d*sum((_R*d+c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=RootOf(_Z^3*b*d^3+
3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))-1/a/d/e^2/(d*x+c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{a d^{2} e^{2} x + a c d e^{2}} - \frac{b \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{a e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*e*x + c*e)^2),x, algorithm="maxima")

[Out]

-1/(a*d^2*e^2*x + a*c*d*e^2) - b*integrate((d*x + c)/(b*d^3*x^3 + 3*b*c*d^2*x^2
+ 3*b*c^2*d*x + b*c^3 + a), x)/(a*e^2)

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Fricas [A]  time = 0.219099, size = 244, normalized size = 1.47 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (d x + c\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} -{\left (a d x + a c\right )} \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3}{\left (d x + c\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b d x + b c + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 6 \,{\left (d x + c\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}} - 2 \, \sqrt{3}{\left (b d x + b c\right )}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 6 \, \sqrt{3}\right )}}{18 \,{\left (a d^{2} e^{2} x + a c d e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*e*x + c*e)^2),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*(d*x + c)*(b/a)^(1/3)*log(b*d^2*x^2 + 2*b*c*d*x + b*c^2 -
(a*d*x + a*c)*(b/a)^(2/3) + a*(b/a)^(1/3)) - 2*sqrt(3)*(d*x + c)*(b/a)^(1/3)*lo
g(b*d*x + b*c + a*(b/a)^(2/3)) - 6*(d*x + c)*(b/a)^(1/3)*arctan(1/3*(sqrt(3)*a*(
b/a)^(2/3) - 2*sqrt(3)*(b*d*x + b*c))/(a*(b/a)^(2/3))) + 6*sqrt(3))/(a*d^2*e^2*x
+ a*c*d*e^2)

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Sympy [A]  time = 3.38947, size = 54, normalized size = 0.33 \[ - \frac{1}{a c d e^{2} + a d^{2} e^{2} x} + \frac{\operatorname{RootSum}{\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} a^{3} + b c}{b d} \right )} \right )\right )}}{d e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*e*x+c*e)**2/(a+b*(d*x+c)**3),x)

[Out]

-1/(a*c*d*e**2 + a*d**2*e**2*x) + RootSum(27*_t**3*a**4 - b, Lambda(_t, _t*log(x
+ (9*_t**2*a**3 + b*c)/(b*d))))/(d*e**2)

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GIAC/XCAS [A]  time = 0.234793, size = 305, normalized size = 1.84 \[ \frac{\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (-2\right )}{\rm ln}\left ({\left | -\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (-2\right )} - \frac{e^{\left (-1\right )}}{{\left (d x e + c e\right )} d} \right |}\right )}{3 \, a} - \frac{\sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (-2\right )} - \frac{2 \, e^{\left (-1\right )}}{{\left (d x e + c e\right )} d}\right )} e^{2}}{3 \, \left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, a^{2} d} - \frac{\left (a^{2} b\right )^{\frac{1}{3}} e^{\left (-2\right )}{\rm ln}\left (\left (\frac{b}{a d^{3}}\right )^{\frac{2}{3}} e^{\left (-4\right )} - \frac{\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} e^{\left (-3\right )}}{{\left (d x e + c e\right )} d} + \frac{e^{\left (-2\right )}}{{\left (d x e + c e\right )}^{2} d^{2}}\right )}{6 \, a^{2} d} - \frac{e^{\left (-1\right )}}{{\left (d x e + c e\right )} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*e*x + c*e)^2),x, algorithm="giac")

[Out]

1/3*(b/(a*d^3))^(1/3)*e^(-2)*ln(abs(-(b/(a*d^3))^(1/3)*e^(-2) - e^(-1)/((d*x*e +
c*e)*d)))/a - 1/3*sqrt(3)*(a^2*b)^(1/3)*arctan(1/3*sqrt(3)*((b/(a*d^3))^(1/3)*e
^(-2) - 2*e^(-1)/((d*x*e + c*e)*d))*e^2/(b/(a*d^3))^(1/3))*e^(-2)/(a^2*d) - 1/6*
(a^2*b)^(1/3)*e^(-2)*ln((b/(a*d^3))^(2/3)*e^(-4) - (b/(a*d^3))^(1/3)*e^(-3)/((d*
x*e + c*e)*d) + e^(-2)/((d*x*e + c*e)^2*d^2))/(a^2*d) - e^(-1)/((d*x*e + c*e)*a*
d)